I am given $N$ observations of pairs of covariates and response $(\mathbf{x}_i, y_i)$. When the response are non-negative integers, by doing Poisson regression I am modelling $y_i \sim \mathrm{Pois}(\mu_i)$ as Poisson random variables with mean $\mu_i$, such that $\ln(\mu_i)$ is a linear function of the covariate $\mathbf{x}_i$. A maximum likelihood estimator for the coefficients of $\mathbf{x}_i$ maximises the Poisson log-likelihood:
$$\sum_{i=1}^N (y_i \ln(\mu_i) - \mu_i)$$
I have seen references to doing Poisson regression with non-negative, non-integers, e.g. How does a Poisson distribution work when modeling continuous data and does it result in information loss?
In this case, what log-likelihood function is used? Do you still use the above function but allow $y_i$ to take the non-integer values?